Cusp Universality for Random Matrices II: The Real Symmetric Case

We prove that the local eigenvalue statistics of real symmetric Wigner-type matrices near the cusp points of the eigenvalue density are universal. Together with the companion paper [arXiv:1809.03971], which proves the same result for the complex Hermitian symmetry class, this completes the last remaining case of the Wigner-Dyson-Mehta universality conjecture after bulk and edge universalities have been established in the last years. We extend the recent Dyson Brownian motion analysis at the edge [arXiv:1712.03881] to the cusp regime using the optimal local law from [arXiv:1809.03971] and the accurate local shape analysis of the density from [arXiv:1506.05095, arXiv:1804.07752]. We also present a PDE-based method to improve the estimate on eigenvalue rigidity via the maximum principle of the heat flow related to the Dyson Brownian motion.Comment: 62 pages. Updated version with additional reference

Eutropelia y Mercadillo en el patio de Mnipodio. Honesto y provechoso entretenimiento en/con la malla prologica de la Novelas Ejemplares Rivista di fascia A per Ispanistica

Se analizan en este trabajo las relaciones entre el prólogo y los demás textos preli­mi­nares de las Novelas ejemplares, evidenciando cómo este sistema de textos breves ha­­bía llegado a ser el escenario en el que se realizaba el difícil equilibrio entre po­der, autor, obra, libreros y público: un sistema que hay que entender como aparato mediador del ac­ceso de los textos a la publicación y al espacio público. A través de ellos se des­cubre un entramado de intereses económicos que amparándose en la mo­ral sirven en realidad para controlar con medios espurios un mercado que es, más bien, también mercadillo de cambalache para descorazonar la competencia y pre­servar privilegios no siempre legítimamente adquiridos. En este panorama, la re­acción que Cer­vantes explica entre líneas en su prólogo a las ejemplares se puede des­cifrar como la búsqueda de una ruta para navegar, de forma honesta y ho­nes­tamente en­tre­tenida, entre otia y negotia, guardando las distancias, en la medida de lo posible, de los varios poderes fácticos o jurídicos, económicos y políticos, que po­dían acechar tal empeño

IST Austria Thesis

In the first part of the thesis we consider Hermitian random matrices. Firstly, we consider sample covariance matrices XX∗ with X having independent identically distributed (i.i.d.) centred entries. We prove a Central Limit Theorem for differences of linear statistics of XX∗ and its minor after removing the first column of X. Secondly, we consider Wigner-type matrices and prove that the eigenvalue statistics near cusp singularities of the limiting density of states are universal and that they form a Pearcey process. Since the limiting eigenvalue distribution admits only square root (edge) and cubic root (cusp) singularities, this concludes the third and last remaining case of the Wigner-Dyson-Mehta universality conjecture. The main technical ingredients are an optimal local law at the cusp, and the proof of the fast relaxation to equilibrium of the Dyson Brownian motion in the cusp regime. In the second part we consider non-Hermitian matrices X with centred i.i.d. entries. We normalise the entries of X to have variance N −1. It is well known that the empirical eigenvalue density converges to the uniform distribution on the unit disk (circular law). In the first project, we prove universality of the local eigenvalue statistics close to the edge of the spectrum. This is the non-Hermitian analogue of the TracyWidom universality at the Hermitian edge. Technically we analyse the evolution of the spectral distribution of X along the Ornstein-Uhlenbeck flow for very long time (up to t = +∞). In the second project, we consider linear statistics of eigenvalues for macroscopic test functions f in the Sobolev space H2+ϵ and prove their convergence to the projection of the Gaussian Free Field on the unit disk. We prove this result for non-Hermitian matrices with real or complex entries. The main technical ingredients are: (i) local law for products of two resolvents at different spectral parameters, (ii) analysis of correlated Dyson Brownian motions. In the third and final part we discuss the mathematically rigorous application of supersymmetric techniques (SUSY ) to give a lower tail estimate of the lowest singular value of X − z, with z ∈ C. More precisely, we use superbosonisation formula to give an integral representation of the resolvent of (X − z)(X − z)∗ which reduces to two and three contour integrals in the complex and real case, respectively. The rigorous analysis of these integrals is quite challenging since simple saddle point analysis cannot be applied (the main contribution comes from a non-trivial manifold). Our result improves classical smoothing inequalities in the regime |z| ≈ 1; this result is essential to prove edge universality for i.i.d. non-Hermitian matrices

From clinical application to cognitive enhancement: the example of methylphenidate

Methylphenidate (MPD) is a central nervous system (CNS) stimulant, which belongs to the phenethylamine group and is mainly used in the treatment of attention deficit hyperactive disorder (ADHD). However, a growing number of young individuals misuse or abuse MPD to sustain attention, enhance intellectual capacity and increase memory. Recently, the use of MPD as a cognitive enhancement substance has received much attention and raised concerns in the literature and academic circles worldwide. The prescribing frequency of the drug has increased sharply asconsequence of the more accurate diagnosis of the ADHD and the popularity of the drug itself due to its beneficial short-term effect. However, careful monitoring is required, because of possible abuse. In this review different aspects concerning the use of MPD have been approached. Data showing its abuse among college students are given, when the drug is prescribed short term beneficial effects and side effects are provided; moreover studies on animal-models suggesting long lasting negative effects on healthy brains are discussed. Finally, emphasis is given to the available formulationsand pharmacology

The Dissipative Spectral Form Factor for I.I.D. Matrices

The Dissipative Spectral Form Factor (DSFF), recently introduced in [arXiv:2103.05001] for the Ginibre ensemble, is a key tool to study universal properties of dissipative quantum systems. In this work we compute the DSFF for a large class of random matrices with real or complex entries up to an intermediate time scale, confirming the predictions from [arXiv:2103.05001]. The analytic formula for the DSFF in the real case was previously unknown. Furthermore, we show that for short times the connected component of the DSFF exhibits a non-universal correction depending on the fourth cumulant of the entries. These results are based on the central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices [arXiv:2002.02438, arXiv:1912.04100]

Entanglement Entropy of Non-Hermitian Eigenstates and the Ginibre Ensemble

Entanglement entropy is a powerful tool in characterizing universal features in quantum many-body systems. In quantum chaotic Hermitian systems, typical eigenstates have near maximal entanglement with very small fluctuations. Here, we show that for Hamiltonians displaying non-Hermitian many-body quantum chaos, modeled by the Ginibre ensemble, the entanglement entropy of typical eigenstates is greatly suppressed. The entropy does not grow with the Hilbert space dimension for sufficiently large systems and the fluctuations are of equal order. We derive the novel entanglement spectrum that has infinite support in the complex plane and strong energy dependence. We provide evidence of universality and similar behavior is found in the non-Hermitian Sachdev-Ye-Kitaev (nSYK) model, indicating the general applicability of the Ginibre ensemble to dissipative many-body quantum chaos.Comment: 5 pages, 4 figures; v2: universality results added, background on Dyson Brownian Motion and alternative definition of von Neumann entropy added to supplemental material, references added, and other minor revisions to match version published in PR